Success probability in Shor's Algorithm

TL;DR

This paper improves the understanding of success probabilities in Shor's algorithm by deriving exact formulas, identifying failure cases, and providing simulation tools for better quantum resource estimation and benchmarking.

Contribution

It presents an improved bound on success probability, exact formulas for failure cases, and a simulation routine for evaluating success probabilities in Shor's algorithm.

Findings

Derived formulas identify all failure cases.

Provided an improved success probability bound.

Developed a simulation routine for evaluation.

Abstract

This paper aims to determine the exact success probability at each step of Shor's algorithm. Although the literature usually provides a lower bound on this probability, we present an improved bound. The derived formulas enable the identification of all failure cases in Shor's algorithm, which correspond to a success probability of zero. A simulation routine is provided to evaluate the theoretical success probability for a given integer when its prime factorization is known with potential applications in quantum resource estimation and algorithm benchmarking.